696 CHINESE OPTICS LETTERS / Vol. 5, No. 12 / December 10, 2007
Study on optical gain of one-dimensional photonic crystals
with active impurity
Zhenghua Li (
)
1,2,4
, Tinggen Shen (
)
2,3
, Xuehua Song (
ÝÝÝ
)
3
,
Junfeng Ma (
)
2,3
, Yong Sheng (
DZDZDZ
)
2,3
, and Gang Wang (
)
3
1
Telecommunication T&R Section, Zhenjiang Watercraft College, Zhenjiang 212003
2
Institute of Applied Physics, Jiangsu University, Zhenjiang 212013
3
Department of Telecommunication Engineering, Jiangsu University, Zhenjiang 212013
4
Department of Electronic Engineering, East China Normal University, Shanghai 200062
Received July 11, 2007
Localized fields in the defect mode of one-dimensional photonic crystals with active impurity are studied
with the help of the theory of spontaneous emission from two-level atoms embedded in photonic crystals.
Numerical simulations demonstrate that the enhancement of stimu lated radiation, as well as the phenomena
of transmissivity larger than unity and the abnormality of group velocity close to the edges of photonic
band gap, are related to the negative imaginary part of the complex effective refractive index of dop ed
layers. This means that the complex effective refractive index has a negative imaginary part, and that the
impurity state with very high quality factor and great state density will occur in the photonic forbidden
band if active impurity is introduced into the defect layer properly. Therefore, the spontaneous emission
can be enhanced, the amplitude of stimulated emission will be very large and it occurs most probably close
to the edges of photonic band gap with the fundamental reason, the group velocity close to the edges of
band gap is very small or abnormal.
OCIS codes: 130.3120, 120.7000, 160.4670, 120.4570, 160.5690, 230.4170.
More and more importance has been attached to the
unique behaviors of electromagnetic wave propagating in
photonic crystals, a novel and artificial photonic mate-
rial with periodic structure, in recent years
[1−4]
.Pho-
tonic band gap, prohibiting electromagnetic waves with
particular frequencies propagating in it, means that sp on-
taneous emission can be suppressed. A solitary transmis-
sion p eak can be achieved by introducing defects into a
photonic band gap structure. It suggests that localized
defect mo des will appear in the band gap if defects are
introduced into the photonic band gap structure
[5]
.
The position and width of transmission peak can also
be controlled to meet our need by doping defect properly
or by adjusting an external voltage because of the me-
dial electro-optic effect. Photonic band gap can suppress
spontaneous emission effectively because the probability
of spontaneous emission is directly proportional to the
state density, while the probability of spontaneous emis-
sion from atoms with optical frequency of spontaneous
emission falling into photonic band gap is nearly zero.
However, if active impurity is introduced into photonic
crystals, the impurity state with very high quality factor
and great state density will occur in the photonic band
gap, so the state density is increased and the correspond-
ing spontaneous emission is enhanced. Study has shown
that by introducing active medium in photonic band gap
materials, there is every probability of amplifying light
with high efficiency, structuring photonic crystal laser
with zero threshold
[6]
.
In this letter, the phenomena that the sum of
reflectivity and transmissivity of a photonic crystal with-
out active impurity is unity and that of a photonic crys-
tal with active impurity is far larger than unity at cer-
tain frequency band are studied by the theory of spon-
taneous emission from two-level atoms
[7]
in detail. In
fact, the interaction between radiation field and matter
is universal, but the interaction can be controlled ac-
tively by adjusting the parameters of photonic crystals
to enha nce stimulated emission extremely. This means
that the doped photonic crystals can be used in dense
wavelength-division multiplexing (DWDM) optical com-
munication systems, thereby plenty of optical amplifiers
can be s aved, and signal capacity and transmission dis-
tance can be increased dramatically.
The model, symbolic system, and its conclusions used
here is taken from Ref. [7]. Here we consider spontaneous
emission from a two-level atom embedded in a photonic
crystal with an upper band, a lower band, and a photonic
gap b etween them. The dispersion relation near the two
band edges could be expressed approximately by
ω
k
=
ω
c1
+ C
1
|k − k
i
10
|
2
(ω
k
>ω
c1
)
ω
c2
− C
2
|k − k
j
20
|
2
(ω
k
<ω
c2
)
, (1)
where ω
c1
and ω
c2
are the cutoff frequencies of the upper
band edge and the lower band edge, respectively, and k
represents both the momentum and polarization of the
modes. The gap width ω
12
= ω
c1
− ω
c2
is assumed far
smaller than ω
c1
, ω
c2
. C
1
and C
2
are model-dependent
constants. k
i
10
and k
j
20
are two finite collections of sym-
metry related points.
The upper level |1 of a two-level atom is coupled by
electromagnetic modes to the lower level |0 with the res-
onant frequency ω
1
between them,
H =¯hω
1
|11| +
k
¯hω
k
b
+
k
b
k
+i¯h
k
g
k
(b
+
k
|01|−b
k
|10|), (2)
1671-7694/2007/120696-04
c
2007 Chinese Optics Letters
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